Alberts was born in Cedar Falls, Iowa to parents Ken Alberts, a business executive, and Linda. He also has an older brother, Troy, who went on to work as a sales director in an oil processor, and a sister, Tami, who teaches elementary school. Trev Alberts attended Northern University High School in Cedar Falls, where he played for the Northern University Panthers high school football team.
While attending the University of Nebraska, he played for the Nebraska Cornhuskers football team from 1990 to 1993. Following his senior season in 1993, he was awarded the Dick Butkus Award and Jack Lambert Trophy as the top college linebacker; Alberts was also recognized as a consensus first-team All-American, after recording 15 quarterback sacks, 21 tackles for loss, and 38 quarterback hurries. Despite an injury early in the eleventh game of the season against the Oklahoma Sooners, Alberts returned with a cast on his arm for the national championship game against Florida State in the Orange Bowl. Although the Seminoles won 18-16, Alberts had a dominant performance with three sacks of Heisman Trophy-winner Charlie Ward (FSU quarterbacks were sacked only five times during the 1993 regular season).
Selected fifth overall in the 1994 draft, Alberts began his professional career with the Indianapolis Colts and continued to play for the Colts from 1994 to 1996. Due to injuries he played portions of just three seasons before retiring before the 1997 season, tallying just four career quarterback sacks and one interception. He is considered one of the fifty biggest draft busts of all time by CBS.
Upon retirement from the NFL, Alberts was hired by the American cable television network CNN/SI and concomitantly its Sports Illustrated magazine, where he served as a college football contributor. In 2002, Alberts joined the staff of the American cable television network ESPN, where he worked as an in-studio analyst for college football, ultimately joining Rece Davis and Mark May on the network's College GameDay Scoreboard and College GameDay Final.
On September 6, 2005, Alberts was terminated by ESPN for breaching his contract when he declined to report to work; Alberts later claimed he did not want to "play second fiddle" to the more prominent cast of College GameDay, Chris Fowler, Kirk Herbstreit and Lee Corso.
Alberts thereafter accepted a position as a columnist for the website of the college sports cable television network CSTV. He worked as a color commentator for the NFL on Westwood One Sunday afternoon radio broadcasts in 2006. He also provided color commentary for SEC football games on CBS.
Alberts also served as an analyst for Sprint Exclusive Entertainment, breaking down college football and other sports for viewers.
Alberts was hired in April 2009 to be the director of athletics for the Nebraska–Omaha Mavericks sports program at the University of Nebraska-Omaha.
Alberts made the controversial decision to eliminate football and wrestling in an effort to bring University of Nebraska-Omaha to Division I's Summit League. The regents approved the move March 25, 2011.